Locked Bag 1797 Penrith NSW 2751 Australia Library

Disclaimer Please note that this is an electronic, post-print version of this article produced by The University of Western Sydney, in accordance with the requirements of the publisher. Whilst this version of the article incorporates refereed changes and has been accepted for publication, difference may exist between this and the final published version. Citations should only be made from the published version. Every reasonable effort has been made to ensure that permission has been obtained for items included in ResearchDirect. If you believe that your rights have been infringed by this repository, please contact Lib-Research@uws.edu.au

1  

Flow induced vibration of two rigidly coupled circular cylinders in tandem and side-by-side arrangements at a low Reynolds number of 150 Ming Zhao

School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith, NSW 2751, AUSTRALIA Tel: +61 2 4736 0085, Email: m.zhao@uws.edu.au

Abstract

Flow induced vibration (FIV) of two rigidly coupled identical circular cylinders in tandem and side-

by-side arrangements at a low Reynolds number of 150 is studied numerically. The two cylinders

vibrate in the cross-flow direction and have the same displacement. The Navier-Stokes equations are

solved by the finite element method and the equation of motion of the cylinders is solved by the

fourth-order Runge-Kutta algorithm. Simulations are conducted for a constant mass ratio of 2 and the

gap ratios (defined as the ratio of the centre-to-centre distance between the two cylinders L to the

cylider diameter D) of 1.5, 2, 4 and 6. The reduced velocities range from 0.5 to 15 with an increment

of 0.5 for the tandem arrangement and 0.5 to 30 with an increment of 0.5 for the side-by-side

arrangement. It is found that the gap between the two cylinders has significant effect on the response.

For a tandem arrangement, the lock-in regime of the reduced velocity is narrower than that of a

single cylinder for L/D=1.5 and 2 and wider than later for L/D=4 and 6. If the two cylinders are

allowed to vibrate in the cross-flow direction, the vortex shedding from the upstream cylinder occurs

at L/D as small as 2. The most interesting phenomenon found in the side-by-side arrangement is the

combination of vortex-induced vibration (VIV) and galloping at L/D=1.5 and 2. For L/D=1.5 and 2,

the response is dominated by VIV as Vr<15 and by galloping as Vr>15. At reduced velocities close to

15, the response is a combination of VIV and galloping.

1. Introduction

Vortex-induced vibration (VIV) of a circular cylinder in fluid flow has been studied extensively

because of its engineering importance. Most of the studies were focused on the fundamental case of

2  

VIV of an elastically mounted rigid circular cylinder in steady flow. When an elastically mounted

cylinder is placed in a fluid flow, large-amplitude oscillations occur when the vortex shedding

frequency synchronizes with the oscillation frequency [1]. The synchronization between the vortex

shedding and the vibration of the cylinder is also called the ‘lock-in’ or ‘lock-on’ in literature. Lock-

in occurs in a range of reduced velocity and the lock-in regime of the reduced velocity is dependent

on the mass ratio and damping ratio of the system [2, 3]. The mass ratio is defined as the ratio of the

cylinder mass to the displaced fluid mass. The reduced velocity Vr is defined as DfUV nr / , where

U is the free-stream velocity, fn is the structural natural frequency and D is the cylinder diameter. The

natural frequency of free vibration of a cylinder in still water (defined as fnw in this study) is used in

many laboratory studies of VIV to define the reduced velocity, i.e. DfUV nwr / [4, 5, 6]. In most of

the numerical studies of VIV at low Reynolds numbers, the reduced velocity is defined using the

structural natural frequency [7, 8]. At high mass ratios, the structural natural frequency and the

natural frequency in fluid are essentially the same and the VIV frequency lock-in with the natural

frequency [9, 10]. When the mass ratio is in the order of O (1), the response frequency can be

significantly greater than the natural frequency measured in still water [6]. Williamson and Roshko

[1] found that the vortex-shedding pattern in the wake of a vibrating cylinder is related to the

response mode of the cylinder. The number of the vortices in each cycle of vibration was used to

define the vortex shedding mode. For example, the 2P mode stands for two pairs of vortices being

generated in one cycle of vibration and 2S mode stands for two single vortices being generated in

one cycle of vibration. For VIV of a circular cylinder in the cross-flow direction at low mass ratios,

the vortex shedding is in the 2S mode in the initial branch and 2P mode in the upper and lower

branch [6]. If a cylinder is allowed to vibrate both in the in-line and the cross-flow directions, the

maximum response amplitude is increased to about 1.5 diameter of the cylinder and the velocity

regime where the amplitude reaches its maximum is defined as super upper branch [5]. 2T mode

3  

(two triplex vortices are shed from the cylinder in each vibration cycle) was observed when the

amplitude is the maximum.

Two circular cylinders close to each other in fluid flow have also attracted much interest. The

interference between the two cylinders has significant effect on the wake vortex shedding flow [11,

12, 13]. Williamson [14] found that for a certain range of gap between two cylinders in a side-by-

side arrangement, the wakes form the two cylinders were synchronized, either in phase or in anti-

phase with each other. When the distance between the centres of two side-by-side cylinder is less

than 2.2D, only one wake is formed [14-17]. Alam et al. [17] found that, as the spacing between two

side-by-side circular cylinders is less than 1.2 diameters, the gap flow was biased toward one side,

resulting in the formulation of a narrow wake behind one cylinder and a wide wake behind the other.

For two cylinders in a tandem arrangement, the vortex shedding is found to occur only from the

downstream cylinder if the distance between the two cylinders is less than a critical distance [18, 19,

20]. This critical distance is about 3.0 – 3.5 diameter depending on the Reynolds number.

VIV of two circular cylinders is not studied as extensively as flow past two stationary cylinders

or VIV of a single cylinder. Zdravkovich [21] provided a review of the effects of the interference

between two cylinders on the VIV. Fontaine et al. [22] classified the response of two riser pipes in a

tandem arrangement in two categories: Wake Induced Oscillation (WIO) and Vortex Induced

Vibration (VIV). WIO, also referred to as galloping, a classical type of instability for a simplified

description of the mechanism underlining the phenomenon and VIV is known to be a self-limited

motion with a maximum order of 1 diameter for a single riser [22]. The galloping in the case of two

cylinders in tandem is also referred to be “interference galloping” [23] or “wake galloping” [24].

Bokaian and Geoola [25] studied vibration of an elastically mounted circular cylinder in the wake of

a fixed upstream cylinder and found that the response of the cylinder can be a VIV, a galloping or a

combination of VIV and galloping, depending on the gap between the two cylinders. Numerical

studies on VIV of two cylinders in tandem were mainly conducted at relatively low Reynolds

4  

numbers and gaps between the two cylinders of 5 or 5.5 diameters [26, 27, 28]. Zang et al. [29] and

Zhao and Yan [30] investigated VIV of two cylinders of different diameter close to each other and

found the interference between the two cylinders has significant effect on the VIV.

The studies of VIV of two side-by-side cylinders in fluid flow are fewer than those of two

tandem cylinders. Huera-Huarte and Gharib [31] investigated vortex-induced vibration of two

flexible cylinders in side-by-side and tandem arrangements. It was found that the interference

between the two cylinders was very weak if the centre-to-centre gap exceeded 3.5 times the cylinder

diameter in the side-by-side configuration. Wang et al. [32] focused on their study on the effects of

turbulence on the response of two side-by-side cylinders in flow. It was found that the enhancement

of the response by the turbulence is significant when the two cylinders are very close to each other.

Fig. 1 Sketch of flow past two cylinders in (a) tandem and (b) side-by-side arrangements

In this study, VIV of two rigidly coupled cylinders in tandem and side-by-side arrangements is

studied by numerical method. Since the two cylinders are rigidly coupled, they vibrate as a single

body. In offshore oil and gas engineering, two pipelines or riser pipes are sometimes strapped

together at a certain intervals, forming a rigidly coupled two-pipe system. In most engineering

applications of cylindrical structures in fluid flow, the Reynolds numbers are in the three-

dimensional turbulent regime. The two-dimensional model at a low Reynolds number is used in this

study based on the following considerations. Firstly, two-dimensional simulations allow extensive

x

y

O

L

DDFlow

(a)

Cylinder 1 Cylinder 2

Cylinder 1

x

y

OL

D

D

Flow

(b)Cylinder 2

5  

numerical simulations under various flow conditions to be conducted at affordable computational

time. Secondly, two-dimensional simulations at laminar flow regime can identify the effects of the

gap between the two cylinders and the flow directions on the response without any influence from

the three-dimensionality. The outcomes found in this study can be guidance for the future three-

dimensional simulations. Two-dimensional (2D) Navier-Stokes equations are solved to simulate the

flow at a Reynolds number of 150. The 2D model is appropriate for the Reynolds number of 150

because the wake flow at this Reynolds number of 150 is two-dimensional and laminar [33, 34].

2. Governing equations and numerical method

Fig. 1 shows the sketch of two rigidly coupled cylinders in tandem and side-by-side

arrangements. The two cylinder system is elastically mounted with zero structural damping. The

cylinder system is allowed to vibrate in the cross-flow direction only. The structural damping is

assigned to be zero to obtain the most intense vibrations [35]. The two cylinders vibrate as a whole

and have same displacement and frequency. The two-dimensional incompressible Navier-Stokes

(NS) equations are the governing equations for simulating the fluid flow. The Arbitrary Lagrangian

Eulerian (ALE) scheme is applied to deal with the moving boundaries of the cylinder surface. The

non-dimensionalization method of the NS equations is the same as that used by Zhao et al. [8]. The

velocity (u, v), the time t, the length (x, y) and the pressure p are nondimensionalized as

)/()~,~(),( Dfvuvu n , nftt ~ , Dyxyx /)~,~(),( , )/(~ 22Dfpp n , respectively, where the tilde

denotes the dimensional parameters, ρ is the fluid density and fn is the structural natural frequency of

the two cylinder system. The non-dimensional ALE formulation of the NS equations for

incompressible flows are expressed as

jj

ir

ij

jjj

i

xx

uV

x

p

x

uuu

t

u

2

mesh, Re, (1)

6  

0

i

i

x

u, (2)

where x1=x and x2=y are the Cartesian coordinates in the in-line and the transverse directions of the

flow, respectively; ui is the fluid velocity component in the xi-direction; t is the time and mesh,iu is the

velocity of the movement of the mesh nodes. The nondimensionalization method used in this study

leads to a nondimensional free-stream velocity that is equal to the reduced velocity and a

nondimensional vibration frequency that is equal to the ratio of the vibration frequency to the

structural natural frequency. The nondimensional equation of the motion of the cylinder is

*

22

2

2 244

m

CVY

t

Y

t

Y Lr

(3)

where Y is the displacement of the cylinder system in the cross-flow direction, dmmm /* is the

mass ratio with m being the mass of the cylinder and md being the displaced mass of the fluid;

kmc 2/ is the damping ratio with c and k being the damping constant and spring constant of

the system, respectively; CL is the lift coefficient of the cylinder system defined by

)2//( 2DUFC LL with LF being the sum of the lift forces on the two cylinders. The equations of

motion Eq. (3) is solved using the fourth-order Runge-Kutta algorithm.

The two-dimensional Navier-Stokes equations are solved by the Petrov-Galerkin Finite Element

Method (PG-FEM) [36]. After each computational time step in the numerical simulation, the

positions of the two cylinders change and each finite element node needs to be moved accordingly.

The governing equation for calculating the displacements of the nodes of the FEM mesh is

0 yS , (4)

where, yS represents the displacement of the nodal points in the y-direction; γ is a parameter that

controls the mesh deformation. In order to avoid excessive deformation of the near-wall elements,

the parameter γ in an finite element is set to be A/1 , with A being the area of the element [36,

37]. By specifying the displacements at all boundaries, Eq. (4) is solved by a Galerkin FEM. The

7  

displacement Sy is the same as the displacement of the cylinders on the cylinder surfaces and zero on

the rest of the boundaries.

Fig. 2 Computational mesh near the two cylinders

A nondimensional rectangular computational domain of a length of 50 (in the flow direction)

and a width of 40 (in the cross-flow direction) is discretized into quadrilateral four-node linear finite

elements. Fig. 2 shows examples of finite elements around the cylinders. Zhao et al. [38] found that

if the width of the computational domain (W) to the cylinder diameter ratio (W/D) is greater than 10

diameter, the effects of the side boundaries on the results has been negligibly weak. In this study,

W/D=80 for a side-by-side arrangement and 40 for a tandem arrangement. The centre of the two

cylinder system is located 20 from the inlet boundary and equal distance to the two lateral boundaries

of the computational domain. Initially the velocity and the pressure in the whole computational

domain are zero. The velocity at the inlet boundary is Vr and the velocity at the cylinder surfaces

equals to the vibration speed of the cylinder. At the outflow boundary the pressure is zero and the

gradient of the velocity in the flow direction is zero. At the two lateral boundaries, the gradient of the

pressure and the velocity in the cross-flow direction are zero.

3. Validation of the numerical model

8  

The numerical model has been validated by Zhao et al. [8] and used to simulate VIV of a square

cylinder in steady flow at a low Reynolds number of 100. In this study, further validation is

conducted for the cases of flow past two stationary cylinders at Re=200, VIV of a circular cylinder at

Re=150 and VIV of a circular cylinder in the wake of a stationary circular cylinder at Re=150. Flows

past two stationary cylinders in a tandem arrangement at gap ratios of L/D=1.5, 2 and 4 and a

Reynolds number of 200 are simulated and the results are compared with the published data in Table

1. Meneghini et al. [18] simulated flow past two cylinders in a tandem arrangement at Re=200 and

L/D=1.5 to 4. Table 1 shows the comparison of the mean drag coefficient DC , the amplitude of the

lift coefficient ACL and the Strouhal number St. The subscripts 1 and 2 stands for the upstream and

the downstream cylinders, respectively throughout the paper. Only the results for L/D=1.5 and 4 are

available in Borazjani and Sotiropoulos [35]. The ACL1 and ACL2 are defined as the amplitudes of the

lift coefficient of the upstream and the downstream cylinders, respectively. Because ACL1 and ACL2 by

Meneghini et al. [18] are measured from the time histories of the force coefficient in their paper,

there is certain uncertainty in them. The results from the three models in Table 1 for L/D=1.5 are

very close to each other. Flow past a stationary single cylinder at Re=200 is simulated and the mean

drag coefficient and the Strouhal number are DC =1.336 and St=0.194, respectively. The

corresponding values calculated by Meneghini et al. [18] are DC = 1.3 and St=0.196, respectively.

The drag coefficient of the upstream cylinder is positive and that of the downstream cylinder is

negative when L/D is small. When L/D is small, the flow pattern is characterized by two quasi-steady

vortices in the gap region and periodic vortex shedding from the downstream cylinder. The drag

coefficient and the Strouhal number at small L/D are smaller than their corresponding values for a

single cylinder. The oscillation of the lift coefficient of both cylinders increases significantly as L/D

is increased from 2 to 4. The lift coefficients are increased because the vortex shedding from the

upstream cylinder occurs at large L/D. Obvious difference occurs among the three sets of results as

L/D=4. The present results agree will those by Meneghini et al. [18] but have big difference from

9  

those in Ref. [35]. Borazjani and Sotiropoulos [35] stated that the difference occurs because the

spacing ratio L/D=4 is close to the critical spacing where the flow patterns undergo an abrupt change.

Table 1 Comparison of the numerical results of flow past two stationary cylinders in tandem Reference L/D 1DC ACL1 St1 2DC ACL2 St2

Borazjani and Sotiropoulos [35] 1.5

1.112 0.024 0.174 -0.216 0.050 0.174 Meneghini et al. [18] 1.060 0.03 0.167 -0.180 0.060 0.167

Present 1.086 0.025 0.166 -0.198 0.063 0.166 Meneghini et al. [18]

2 1.030 0.070 0.130 -0.170 0.149 0.130

Present 1.046 0.030 0.131 -0.208 0.106 0.130 Borazjani and Sotiropoulos [35]

4 1.290 0.745 0.185 0.600 1.900 0.185

Meneghini et al. [18] 1.180 0.75 0.174 0.380 1.500 0.174 Present 1.269 0.821 0.172 0.366 1.607 0.172

The second validation case is the VIV of a circular cylinder at low Reynolds number of Re=150,

low reduced mass of Mred=2 and zero damping ratio. It should be noted that the reduced mass, which

is defined as 2red / DmM , is different from the mass ratio used in this study. In this example

2red M is used in order to compare the numerical results with other published data under the same

calculation parameters. Fig. 3 (a) shows the comparison between the calculated response amplitudes

with the numerical results in three published papers. The amplitude is defined as 2/)( minmax YYAy

. The variations of the response amplitude with the reduced velocity from different models are

similar to each other. The present results are closer to the results by Bao et al. [39]. Fig. 3 (b) shows

the comparison of the calculated mean drag coefficient, root mean square (RMS) drag and lift

coefficients with their counterparts calculated by Bao et al. [39]. The agreement between the two sets

of results are very good except the mean drag coefficient at Vr=4, where the vibration amplitude is

the maximum. In the present results, all the mean drag, RMS drag and RMS lift coefficients, and the

response amplitude reach their maximum values at Vr=4. The numerical study by Singh and Mittal

[7] of 2DOF VIV and the experimental study by Khalak and Williamson [4] of 1DOF VIV in the

cross-flow direction also showed that the mean drag coefficient, the RMS drag and lift coefficients

10  

reached their maximum values at a same reduced velocity. In the results by Bao et al. [39], the

response amplitude and the lift coefficient at Vr=4 are the maximums, while the mean drag

coefficient and the RMS drag coefficient at Vr=4 are not.

The validation of the numerical model on the interference between two cylinders on VIV is

testified by comparing the numerical results with those by Bao et al. [39], who studied flow

interference between a stationary cylinder and an elastically mounted cylinder arranged in a tandem

arrangement. Vibration of an elastically mounted circular in the wake of an identical stationary

cylinder at Re=150 is simulated. The gap ratio L/D, the reduced mass and the damping ratio of the

downstream cylinder is the same as those used by Bao et al. [39], i.e. L/D=4, Mred=2 and ζ=0. Fig. 4

shows the variations of the response amplitude and the root mean square lift coefficient with the

reduced velocity. The experimental data by Assi et al. [41] are also plotted in Fig. 4 (a) for

comparison. In the experiments, the mass ratio was 2 (equivalent to Mred=1.57), which is slightly

lower than that used in the numerical simulation and the Reynolds numbers are in the turbulent flow

regime in range of 3000 – 13000. The present results of both response amplitude and the lift

coefficient agree exceptionally well with their counterparts calculated in Ref. [39]. The lock-in

regime of the response for the two cylinder system in Fig. 4 is much wider than that of a single

cylinder. Both numerical results agree well with the experimental data before the response amplitude

reaches its maximum (Vr<6). As Vr>6, the response amplitude in the experiments continues

increasing with the increasing Vr while that from the numerical simulation decreases. The

discrepancy between the experimental data and the numerical results is attributed to the difference in

the Reynolds number.

11  

Fig. 3 Variations of the response amplitude and the force coefficients with the reduced velocity for a single cylinder at Re=150, Mred=2 and ζ=0.

Fig. 4 The response amplitude and the root mean square lift coefficient of an elastically mounted

cylinder in the wake of stationary cylinder with L/D=4, Mred=2 and ζ=0

0

0.2

0.4

0.6

0.8

1

2 3 4 5 6 7 8 9

Ay/

D

Ahn and Kallinderis [40]Borazjani and Sotiropoulos [35]Bao et al. [39]Present

(a) Response amplitude

0

0.5

1

1.5

2

2.5

3

2 3 4 5 6 7 8 9

For

ce c

oeff

icie

nt

Vr

Mean CD, Bao et al. [39]Mean CD, PresentRMS CL, Bao et al. [39]RMS CL, PresentRMS CD, Bao et al. [39]RMS CD, Present

(b) Force coefficient

0

0.2

0.4

0.6

0.8

1

1.2

1.4

3 4 5 6 7 8 9 10 11 12

Ay/

D

Vr

Bao et al. [39]PresentAssi et al. [41]

(a) Response amplitude of the downstreamcylinder

m*≈2, m*ζ≈3, Re=3000 - 13000

0

0.5

1

1.5

3 4 5 6 7 8 9 10 11 12

CL

,rm

s

Vr

Bao et al. [39]

Present

(b) Root mean square of the lift coefficientof the downstream cylinder

12  

4. VIV of two cylinders in the tandem arrangement

VIV of two rigidly coupled circular cylinders in the tandem arrangement at gap ratios of

L/D=1.5, 2, 4 and 6 and reduced velocities in the range of 0.5 to 15 with an increment of 0.5 are

simulated. Flow past two stationary cylinders in the tandem arrangement at Re=150 is firstly

simulated and the variations of the mean drag ( DC ) and the root mean square (RMS) lift coefficients

of both cylinders with the gap ratio L/D are shown in Fig. 5. The variation of mean drag coefficient

DC of the downstream cylinder with L/D by Zdravkovich and Pridden [42] is plotted in Fig. 5 (a) for

comparison. The mean drag coefficient on the downstream cylinder is negative at L/D=1.5 and 2,

because vortex shedding from the upstream cylinder does not occur in such small gaps. At small

gaps, the shear layers that are separated from the two sides of the upstream cylinder reattach to the

downstream cylinder, instead of forming vortex shedding flow in the gap between the two cylinders.

Menghini et al. [18] reported that the force could be negative at small gap ratios because the

downstream cylinder was inside the wake formed by the upstream cylinder where the pressure is

low. Zdravkovich and Pridden [42] measured the pressure for flow past two cylinders in tandem and

confirmed that the pressure in front of the downstream cylinder was lower than that downstream at

low gap ratios, leading to a negative drag force. As the gap ratio is increased to L/D=4, the vortex

shedding from the upstream cylinder occurs. The difference between the mean drag coefficient of the

upstream cylinder and that of a single cylinder is small. The RMS lift coefficient on both cylinders

are extremely weak at L/D=1.5 and 2 because of two reasons. Firstly the vortex shedding does not

occur in the gap between the two cylinders at small gap ratios. Secondly, the location, where

interaction between the free shear layers from the two sides of the downstream cylinder occurs, is far

downstream of the cylinders, making very small influence on the lift forces on the two cylinders.

Once L/D is 4, the RMS lift coefficient on the downstream cylinder is significantly greater than that

of a single cylinder, while the RMS lift coefficient of the upstream cylinder is almost the same as

13  

that of a single cylinder. The RMS lift coefficient is greater than that of a single cylinder at large gap

ratios because of the influence of the unsteady vortex shedding flow from the upstream cylinder.

Zdravkovich and Pridden [42] found that the flow changes modes at about L/D=3.5. When L/D<3.5,

there is almost no gap flow and the drag on the downstream cylinder is negative. As L/D>3.5, the

vortex shedding from the upstream cylinder is almost fully developed and impinges on the

downstream cylinder. Meneghini et al. [18] found that the critical L/D between the two flow patterns

is L/D=3 for a low Reynolds number of Re=200.

Fig. 5 Variations of the mean drag and root mean square lift coefficients with L/D for two stationary

cylinders in the tandem arrangement

Fig. 6 shows the time histories of the vibration displacement of the two cylinders in the tandem

arrangement. At L/D=1.5, the vibrations of all the reduced velocities are perfectly periodic. The

amplitude reaches its maximum at Vr=4.5. Beating is the typical phenomenon occurring at Vr=4.5 to

6 for L/D=2. The beating phenomenon was also found in the numerical simulation of VIV of a single

cylinder at reduced velocities where the response amplitude was reaching its maximum [7, 43]. The

beating period increases with an increase in the reduced velocity. In each beating period, the

vibration amplitude increases slowly with time and then quickly reduces to very small amplitude

followed by another beating period. The vibration histories for L/D=4 are regular except with slightly

variation in the amplitude as shown in Fig. 6 (h). For L/D=4, the vibration at Vr=6 shown in Fig. 6

-0.5

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6

CD

L/D

Upstream cylinderDownstream cylinderSingle cylinderExp., downstream cylinder, Re=31000

(a) Mean Drag coefficient

[42]

0

0.5

1

1.5

2

1 2 3 4 5 6

CL

,rm

s

L/D

Upstream cylinderDownstream cylinderSingle cylinder

(b) RMS lift coefficient

14  

(g) has the largest amplitude. For L/D=6, each time histories between 5.139 r V comprises two

frequency components as shows Fig. 6 (k) and (l), with one frequency being about twice another.

Fig. 6 Time histories of the vibration displacement for the tandem arrangement

Fig. 7 shows the amplitude spectra of the vibration for some typical cases, which are obtained

based on the Fast Fourier Transform (FFT). The spectra of the vibration displacement and the lift

coefficient are plotted in the same figure in order to see the link between the vibration and the lift

force. The spectra of the lift coefficient are based on the total lift force, i.e. the sum of the lift forces

on the upstream and downstream cylinders. Fig. 7 (a) and (b) are typical regular periodic vibrations

with single peak frequencies. Fig. 7 (c) and (d) are the spectra corresponding to the beating shown in

Fig. 6 (d) and (e), respectively. In Fig. 7 (c) the spectrum comprises a number of peaks and the

secondary peak frequency is very close to the primary peak frequency (the peak frequency with the

-0.4

-0.2

0

0.2

0.4

250 260 270 280 290 300 310 320 330 340 350

Y/D

(a) L/D=1.5,Vr=4.5

-0.4

-0.2

0

0.2

0.4

250 260 270 280 290 300 310 320 330 340 350

Y/D

(b) L/D=1.5,Vr=5

-1

-0.5

0

0.5

1

250 260 270 280 290 300 310 320 330 340 350

Y/D

(c) L/D=2,Vr=4

-1

-0.5

0

0.5

1

200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

Y/D

(d) L/D=2,Vr=5

-1

-0.5

0

0.5

1

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

Y/D

(e) L/D=2,Vr=6

-1

-0.5

0

0.5

1

200 210 220 230 240 250 260 270 280 290 300Y/

D

(f) L/D=2,Vr=6.5

-1

-0.5

0

0.5

1

80 90 100 110 120 130 140 150 160 170 180

Y/D

(g) L/D=4,Vr=6

-1

-0.5

0

0.5

1

80 90 100 110 120 130 140 150 160 170 180

Y/D

(h) L/D=4,Vr=7.5

-1

-0.5

0

0.5

1

80 90 100 110 120 130 140 150 160 170 180

Y/D

(i) L/D=6,Vr=6

-1

-0.5

0

0.5

1

40 50 60 70 80 90 100 110 120 130 140

Y/D

(j) L/D=6,Vr=7.5

-0.4

-0.2

0

0.2

0.4

60 70 80 90 100 110 120

Y/D

tfn

(k) L/D=6,Vr=9

-0.4

-0.2

0

0.2

0.4

50 60 70 80 90 100 110

Y/D

tfn

(l) L/D=6,Vr=10

15  

highest amplitude). In Fig. 7 (d), the peak frequencies of the displacement and the lift coefficient are

not as distinct as those in Fig. 7 (c). The energy at the frequencies close to the primary peak

frequency leads to the beating. Each spectrum in Fig. 7 (e) and (f) has two peaks and the amplitudes

corresponding to these two peaks are comparable with each other. The lift coefficient corresponding

to the lower peak frequency is smaller than that corresponding to the higher peak frequency.

However, the lower peak frequency has higher amplitudes because it is closer to the natural

frequency. All the spectra shown in Fig. 7 are in the lock-in regime, where the vortex shedding

frequency locks onto the vibration frequency. The frequency of the highest peak in the displacement

spectrum is the same as that in the lift coefficient spectrum except in Fig. 7 (e) and (f).

Fig. 7 Amplitude spectra of the vibration based on the FFT analysis

Fig. 8 shows the variation of the response amplitude with the reduced velocity. The results for

the four calculated gap ratios are plotted separately in Fig. 8 (a) and (b) in order to avoid the

unclearness due to too many curves and symbols in one figure. The results for a single cylinder are

also plotted in the figure for comparison. Fig. 9 shows the variation of the response frequency with

the reduced velocity. If the vortex shedding frequency and the vibration frequency follows the

Strouhal law, vibration frequency should be a linear function of the reduced velocity. For a single

cylinder, the vortex shedding frequency locks onto the vibration frequency of the cylinder instead of

following the Strouhal law when synchronization occurs [9]. At high mass ratios in the order of

O(100), the vibration frequency is the same as the natural frequency of the cylinder [6, 44].

0

0.2

0.4

0.6

0.8

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(a) L/D=1.5, Vr=4.5

0

0.2

0.4

0.6

0.8

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(b) L/D=1.5, Vr=5

0

0.5

1

1.5

0

0.1

0.2

0.3

0.4

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(c) L/D=2, Vr=5

0

0.1

0.2

0.3

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(d) L/D=2, Vr=6

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(e) L/D=6, Vr=9

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(f) L/D=6, Vr=10

16  

However, at low mass ratios, the vibration frequency at lock-in was found to be slightly greater than

the natural frequency of the cylinder [4, 6, 45]. The response amplitude in the lock-in regime is

significantly higher than those outside the lock-in regime. It can be seen in Fig. 8 that the response

amplitude for a single cylinder is increased suddenly as the reduced velocity is increased from 3 to

3.5. The vibration frequency as 3r V exactly follows a linear function of the reduced velocity. As

4rV , the response frequency for a single cylinder increases slowly with the increasing Vr until

Vr=6 and keeps almost constant in the regime of 86 r V . The amplitudes in the regime of

85.3 r V are significant greater than those in other reduced velocities. The lock-in regime for a

single cylinder is believed to be 84 r V .

 

Fig. 8 Variation of the response amplitude with the reduced velocity for the tandem arrangement

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ay/

D

Tandem, L/D=1.5

Tandem, L/D=2

Single cylinder

(a) Amplitude for L/D=1.5 and 2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ay/

D

Vr

Tandem, L/D=4

Tandem, L/D=6

Single cylinder

(b) Amplitude for L/D=4 and 6

17  

Fig. 9 Variation of the response frequency with the reduced velocity for the tandem arrangement

It can be seen in Fig. 8 and Fig. 9 that the lock-in regime for two cylinders in tandem is

significantly different from that of a single cylinder and is dependent on the gap ratio L/D. The lock-

in regime is the narrowest at L/D=1.5. The variation of the response frequency f/fn with the reduced

velocity for L/D=1.5 is almost a linear function except at Vr=4.5, 5 and 5.5, where f/fn is slightly

deviated from the linear function. The response amplitude for L/D=1.5 reaches its maximum at

Vr=4.5. As 6r V , the response amplitude for L/D=1.5 has been very small and changes little with

the increasing Vr. Based on Fig. 8 (a) and Fig. 9 (a), the lock-in regime for L/D=2 is 63 r V ,

which is also narrower than that of a single cylinder. The response frequency for L/D=2 in the lock-

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Tandem, L/D=1.5

Tandem, L/D=2

Single cylinder

(a) Frequency for L/D=1.5 and 2

f/fn=1

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Vr

Tandem, L/D=4

Tandem, L/D=6

Single cylinder

(b) Frequency for L/D=4 and 6

f/fn=1

18  

in regime is smaller than the structural natural frequency, i.e. f/fn<1. The maximum response

amplitude in the lock-in regime for L/D=2 is much higher than that for L/D=1.5.

At L/D=4, the clear cut lock-in regime is hard to be identified in Fig. 9 (b) because the vibration

frequency increases smoothly with the increasing Vr. There is not any disconnection in the variation

of the response amplitude with the reduced velocity either. Taking a close look at the frequency v.s.

reduced velocity curve in Fig. 9 (b), it can be seen that the response frequency is a linear functions of

the reduced velocity in the reduced velocity regimes of 5.35.0 r V and 155.8 r V . In the regime

of 84 r V , the increase rate of the response frequency for L/D=4 with the increasing Vr is much

smaller than those in other ranges of Vr. It can be seen in Fig. 8 (b) that the response amplitudes in

the range of 84 r V are high. The regime of 84 r V is treated to be lock-in regime for L/D=4 in

this study. The vibration amplitude as Vr>8 decreases gradually with the increasing Vr and is still

above 0.35 at Vr=15.

In Fig. 8 (b) there are two maximum values in the response amplitude curve for L/D=6. One is at

Vr=4 and another one is at Vr=7. The response frequency is a linear function of the reduced velocity

as 3r V and it increases suddenly as Vr is increased from 3 to 3.5. A sudden decrease in the

response frequency occurs as the reduced velocity is increased from 4.5 to 5. The nondimensional

response frequency f/fn increases from a value slightly less than 1 to a value slightly greater than 1 as

the reduced velocity is increased from 5 to 8.5. The response amplitude for L/D=6 increases with the

increasing Vr until Vr=7 after which the response amplitude almost keeps constant until Vr=8.5. The

response frequency decreases suddenly to a value slightly less than 1 as the reduced velocity

increases from 8.5 to 9 and then increases slowly with the increasing reduced velocity until Vr=13.5,

where f/fn is about 1. Although the response frequency f/fn in the reduced velocity regime of

5.139 r V is close to 1, the response amplitude is less than 0.2 in this region as shown in Fig. 8 (b).

The reason of the low response amplitude at frequencies close to the natural frequency can be

explained by examine the spectra as shown in Fig. 7 (e) and (f), where the total lift coefficient of the

19  

cylinder system has two predominant frequencies with one being close to the natural frequency and

another one being not. The frequency component of the lift coefficient that is close to the natural

frequency is secondary compared with the one with the higher amplitude. The secondary component

of the force generates higher amplitude than the primary component because it synchronizes with the

natural frequency. However, the amplitude Ay/D is still less than 0.2 because force corresponding to

the vibration frequency is the secondary component. The regime of 5.139 r V is not included in

the lock-in regime because the vortex shedding frequency which is the same as the leading frequency

of the lift coefficient is not the same as the vibration frequency of the cylinder system and the

response amplitude in this range of reduced velocity is very small. So, the whole lock-in regime for

L/D=6 is 5.85.3 r V .

Fig. 10 shows the variation of the RMS lift coefficient of each individual cylinder and the RMS

lift coefficient based on the total force (sum of the lift forces on both cylinders) with the reduced

velocity. The lift coefficient based on the total force is defined as the total lift coefficient. The RMS

lift coefficient on the upstream cylinder is generally smaller than that on the downstream cylinder

because the downstream cylinder is influenced by the vortex shedding flow in the gap. It can be seen

by comparing Fig. 8 (a) with Fig. 10 (c) that in the initial branch of the response (where the response

amplitude increases with the increasing Vr), the RMS lift coefficient of the total force also increases

with the increasing Vr as the gap ratio L/D=1.5 and 2. This is similar to that of a single cylinder case

reported in Ref. [4]. For L/D=1.5 and 2, the RMS lift coefficient peaks at the reduced velocity where

the response amplitude peaks. Different from those as L/D=1.5 and 2, the RMS total lift coefficient

for L/D=4 and 6 reaches its minimum value when the response amplitude peaks. The variation of

RMS total lift coefficient with the reduced velocity as L/D=4 and 6 is similar to the case where the

upstream cylinder is fixed and the downstream cylinder is free to vibrate (as shown in Fig. 4),

because in these two gaps, the vortex shedding from the gap has significant effects on VIV. The total

lift coefficient at the highest calculated reduced velocity for L/D=3 is significantly greater than those

20  

at other gaps. This leads to big response amplitudes even if the lock-in does not occur as shown in

Fig. 8 (b).

Fig. 10 Variation of the RMS lift coefficient with the reduced velocity for two cylinders in the

tandem arrangement

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Tandem, L/D=1.5

Tandem, L/D=2

Tandem, L/D=4

Tandem, L/D=6

(a) CL,rms of the upstream cylinder

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Tandem, L/D=1.5

Tandem, L/D=2

Tandem, L/D=4

Tandem, L/D=6

(b) CL,rms of the downstream cylinder

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Vr

Tandem, L/D=1.5

Tandem, L/D=2

Tandem, L/D=4

Tandem, L/D=6

(c) CL,rms of the total force

21  

Fig. 11 Variation of the phase between the response displacement and the total lift force with the

reduced velocity for tandem arrangement  

Fig. 11 shows the variation of the phase between the response displacement and the total lift

force (ψ) with the reduced velocity for the tandem arrangement. It was found that, for a single

cylinder, the phase between the response displacement and the lift force jump from 0° to 180° as the

response transits from upper branch to lower branch [4]. Similar to that of the single cylinder case,

the phase ψ jumps from 0° to 180° at Vr=6, 6.5 and 6.5 for L/D=1.5, 2 and 4, respectively. It can be

seen by comparing Fig. 8 and Fig. 11 that the jump of the phase occurs after the response amplitude

reaches its maximum value. It is interesting to see that the phase ψ for L/D=6 jumps to 180° at

Vr=7.5 and jumps back to 0° at Vr=9. It jumps from 0° to 180° again at Vr=12.5. The phase ψ

becomes 0° again in the range of 129 r V , where the response frequency approaching natural

frequency as seen in Fig. 9 (b). In the reduced velocity range of 129 r V , both the response

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Tandem, L/D=1.5

Tandem, L/D=2

(c) Phase between Y and CL for L/D=1.5 and 2

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Vr

Tandem, L/D=4

Tandem, L/D=6

(c) Phase between Y and CL for L/D=4 and 6

22  

displacement and the lift force have double frequencies as shown in Fig. 7 (e) and (f), with the lower

frequency being closing to the natural frequency. It is found that phase ψ between the response

displacement and the lift force corresponding to the lower frequency is about 0° and that

corresponding to the higher frequency is about 180°. Fig. 11 (b) only shows the phase corresponding

to the lower frequency for L/D=6 and 129 r V since the response amplitude at this frequency is

higher than another. Because the lift coefficient is related to the vortex shedding, the phase between

the response displacement and the lift coefficient reflects the relation between the response and the

vortex shedding flow. For example, it is found that if the response displacement is in phase with the

lift coefficient, whenever the cylinders reach their lowest position, the vortices below the cylinders

are stronger than those above the cylinders, leading to negative lift coefficient.

Fig. 12 Vorticity contours around two cylinders in a tandem arrangement at L/D=1.5 and Vr=4.5

The vortex shedding flow pattern is examined by observing the contours of the vorticity around

the cylinders. The vorticity is defined as yuxv // . Fig. 12 shows the vorticity contours

around two cylinders in the tandem arrangement at L/D=1.5 and Vr=4.5, where the amplitude is the

maximum. The four instants for the vortex shedding flow structures are marked in the time histories

-1.5

-1

-0.5

0

0.5

1

1.5

-0.4

-0.2

0

0.2

0.4

528 528.5 529 529.5 530 530.5 531

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

23  

of the cylinder displacement and the lift coefficients, which are shown at the bottom of Fig. 12.

Periodic vortex shedding is observed and two vortices are found to be shed in one cycle of vibration.

The vortex shedding pattern in Fig. 12 is very similar to that of two stationary cylinders in tandem at

the same gap ratio. The vortices are only shed from the downstream cylinder, while the fluid velocity

in the gap between the two cylinders is very weak. The two cylinders behave as a single body which

is more streamlined. The hydrodynamic lift coefficient on the upstream cylinder is very small since

there is not vortex shedding behind it. The total lift coefficient 2L1L CC is in-phase with the

vibration displacement, resulting large response amplitude.

Fig. 13 Vorticity contours for two cylinders in a tandem arrangement at L/D=2 and Vr=5

-4

-3

-2

-1

0

1

2

3

4

-1

-0.5

0

0.5

1

360 361 362 363 364 365 366 367 368 369

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a) (b) (c) (d) (e) (f)

24  

Fig. 13 shows the vorticity contours around two cylinders in the tandem arrangement at L/D=2

and Vr=5. For two stationary cylinders in tandem, it has been found that the vortex shedding from the

upstream cylinders does not occurs if the gap ratio L/D is less than about 3.5 or 4 [42, 18]. The

critical gap for the onset of the vortex shedding from the upstream cylinder at low Reynolds numbers

is slightly greater than that at high Reynolds numbers. At L/D=2, the vortex shedding does not occur

behind the upstream cylinder if the two cylinders are stationary. This can be evidenced by the

extremely small RMS lift coefficients on both cylinders as shown in Fig. 5 (b). However, it can be

been in Fig. 13 that the vortex shedding behind the upstream cylinders occurs at a gap ratio as small

as L/D=2. In Fig. 13 (b), where the cylinders are moving downwards, the negative vortex from the

top side of the upstream cylinder separates from the cylinder surface. This vortex is shed from the

cylinder because the relative direction of the fluid flow to the cylinders is biased upwards due to the

downward movement of the cylinders. This allows the vortex from the upstream cylinder being shed

from the top-right direction. The negative vortex that is shed in Fig. 13 (b) scratches the top side of

the downstream cylinder as seen in Fig. 13 (c). The lift force on the downstream cylinder is very

small because the negative vortex coming from the top side of the upstream cylinder cancels the

downward forces by the positive vortex from the bottom of the downstream cylinder. It appears that

the hydrodynamic force on the upstream cylinder dominate the vibration. For L/D=2, the vortex

shedding from the upstream cylinder is only observed in the lock-in regime, where the response

amplitude is large. Outside the lock-in regime, the vortex shedding only occurs in the wake of the

downstream cylinder and the fluid velocity in the gap is very small, which is similar to that in Fig.

12. The beating, which is a typical phenomenon for L/D=2 and Vr=5 as shown in Fig. 6 (d), is found

to be related to the change in the vortex shedding. It is found that the vortex shedding from the

upstream cylinder disappear and reappear periodically. Whenever the vortex shedding from the

upstream cylinder is suppressed, the response amplitude is very small as shown in Fig. 13 (e) and (f).

25  

The period of beating in the time history shown in Fig. 6 (d) is the same as the period of the

disappearance and reappearance of the vortex shedding from the upstream cylinder.

Fig. 14 shows Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and Vr=6,

where the vortex shedding from the upstream cylinder is fully developed. After each vortex is shed

from the upstream cylinder, it impinges on the downstream cylinder and affects the vortex shedding

from the downstream cylinder significantly. Fig. 14 (a) and (c) shows that each vortex from the

upstream cylinder is split into two parts by the downstream cylinder. The lift coefficients on the two

cylinders are almost in anti-phase with each other. In Fig. 14 (b) the lift coefficient on the upstream

cylinder is positive because the stagnation point of the flow is located on the bottom-left side of the

upstream cylinder and the flow relative to the cylinder is in the top-right direction. However, the lift

coefficient on the downstream cylinder in Fig. 14 (b) is negative because of attraction of the strong

positive vortex attaching to the bottom surface of the cylinder. After the positive vortex that is shed

from the upstream cylinder in Fig. 14 (a) attacks the downstream cylinder, most of this vortex is

covected to the bottom side of the downstream cylinder and resulting strong positive vortex from at

the bottom surface of the downstream cylinder in Fig. 14 (b).

26  

Fig. 14 Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and Vr=6

The vortex shedding pattern is found affected not only by the gap ratio L/D but also the reduced

velocity. Fig. 15 shows vorticity contours for two cylinders in a tandem arrangement when the

cylinders are at the lowest location for L/D=4 at four reduced velocities. At Vr=4, each negative

vortex that has been totally separated from the upstream cylinder will be convected by the flow for

half period of time before it reaches the downstream cylinder. It passes through the top side of the

downstream cylinder. With the increase in the reduced velocity, the shear layers from the upstream

cylinder extend further downstream and the vortices attack the downstream cylinder right after they

are shed from the upstream cylinder. The vortex shedding from the upstream cylinder is always in

the 2S mode, i.e. two single vortices are shed from the upstream cylinder in each vibration period.

Singh and Mittal [7] also found that the 2S vortex shedding mode in the wake of an elastically

mounted cylinder at low Reynolds numbers less than 300.

-2

-1

0

1

2

-1.5

-1

-0.5

0

0.5

1

1.5

240.5 241 241.5 242 242.5 243 243.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

27  

Fig. 15 Vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the

lowest location for L/D=4 and Vr=4, 4.5, 5 and 7

Fig. 16 shows the vorticity contours for two cylinders in a tandem arrangement at L/D=6 and

Vr=4. The influence of the vortex shedding from the upstream cylinder on that of the downstream

cylinder is very similar to that in Fig. 15 (a). Each vortex that is shed from the upstream cylinder

reaches the downstream cylinder after about one vortex shedding period. Instead of impinging on the

front surface of the downstream cylinder, the vortices from the upstream cylinder scratch the top and

the bottom surfaces of the downstream cylinder. The two rows of vortices from the upstream

cylinder attract the vortices that are shed from the downstream cylinder upwards and downwards,

respectively. The vortex street downstream the downstream cylinder is the combination of the vortex

shedding from both cylinders. Fig. 17 shows Vorticity contours for two cylinders in a tandem

arrangement at L/D=6 and Vr=8. The vortex shedding in Fig. 17 is similar to that in Fig. 15 (c) and

(d). It can be seen that the vortex shedding in Fig. 17 (a) is not exactly the same as that in Fig. 17 (c).

The positive vortex that has been shed from the upstream cylinder in Fig. 17 (a), where the cylinders

are in their highest position, attacks the front surface of the downstream cylinder. However, in Fig.

17 (c), where the two cylinders are in their lowest position, the negative vortex in the gap has not

reached the downstream cylinder. This negative vortex will pass through the top surface of the

downstream cylinder, instead of attacking the front surface of the downstream cylinder. It is found

28  

that the vortex shedding for Vr<8 is similar to that shown in Fig. 17 (c) and that for Vr>8 is similar to

that shown in Fig. 17 (a).

Fig. 16 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=4

5. VIV of two cylinders in the side-by-side arrangement

Flow past two stationary cylinders in the side-by-side arrangement at Re=150 is firstly simulated

and the calculated mean drag and the RMS lift coefficients are shown in Fig. 18. The two cylinders

are labelled by Cylinders 1 and 2, respectively as shown in Fig. 1. The statistic values of the force

coefficients of Cylinder 1 and those of Cylinder 2 are the same because the symmetric configuration.

The mean drag coefficient is greater than that of a single cylinder due to the increase in the blockage

effect. The mean lift coefficient is very small at L/D=1.5 and is increased to its maximum at L/D=4.

The effect of the interference between the two cylinders is still obvious at L/D=6 judging by the

mean drag and the RMS lift coefficients.

-5-4-3-2-1012345

-1

-0.5

0

0.5

1

325 325.5 326 326.5 327 327.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

29  

Fig. 17 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=8

Fig. 18 Variations of mean drag and root mean square lift coefficients with L/D for two cylinders in

the side-by-side arrangement at Re=150  

Fig. 19 shows the time histories of the displacement of the two cylinders for the side-by-side

arrangement. Alam et al. [17] found that the flow in the wake of two side-by-side cylinders was

unstable if the gap ratio is less than 2.2. In this study, the flow for L/D=1.5 is found to be unstable at

very small and very large reduced velocities (as shown in Fig. 19 (a) and (d)). The amplitude

responses of the cylinders are perfectly periodic as shown in Fig. 19 (b) and (c). For L/D=1.5 the

-2

-1

0

1

2

-1.5

-1

-0.5

0

0.5

1

1.5

132 132.5 133 133.5 134 134.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a) (b) (c) (d)

0

0.5

1

1.5

2

1 2 3 4 5 6

CD

L/D

Cylinder 1

Cylinder 2

Single cylinder

(a) Mean Drag coefficient

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6

CL

,rm

s

L/D

Cylinder 1Cylinder 2Single cylinder

(b) RMS lift coefficient

30  

vibration of the cylinders is extremely irregular as Vr=20 and 30. The very irregular vibrations are

also found at L/D=2 and high reduced velocities as shown in Fig. 19 (i) and (j). For L/D=2, the

vibration is found to be periodic as 13r V and irregular as 14r V . It can be found that the response

of the cylinders at Vr=13.5 is regular sometimes and irregular sometimes. The vibration of the two

cylinders for L/D=4 and 6 are always regular and periodic in the whole range of simulated reduced

velocities.

Fig. 19 Time histories of the cylinder displacement for the side-by-side arrangement

-0.1

0

0.1

350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500

Y/D

(a) L/D=1.5,Vr=2.5

-1

-0.5

0

0.5

1

250 260 270 280 290 300 310 320 330 340 350

Y/D

(b) L/D=1.5,Vr=5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110

Y/D

(c) L/D=1.5,Vr=15.5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110

Y/D

(d) L/D=1.5,Vr=16

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110

Y/D

(e) L/D=1.5,Vr=20

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80

Y/D

(f) L/D=1.5,Vr=30

-1

-0.5

0

0.5

1

200 210 220 230 240 250 260 270 280 290 300

Y/D

(g) L/D=2,Vr=6.5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110 120 130 140

Y/D

(h) L/D=2,Vr=13.5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90

Y/D

(i) L/D=2,Vr=20

-1

-0.5

0

0.5

1

10 20 30 40 50 60

Y/D

(j) L/D=2,Vr=30

-1

-0.5

0

0.5

1

60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220

Y/D

(k) L/D=4,Vr=3.5

-1

-0.5

0

0.5

1

310 320 330 340 350 360 370 380 390 400 410 420

Y/D

(l) L/D=4,Vr=6.5

-1

-0.5

0

0.5

1

160 170 180 190 200 210 220 230 240 250 260

Y/D

tfn

(m) L/D=6,Vr=4

-1

-0.5

0

0.5

1

190 200 210 220 230 240 250 260

Y/D

tfn

(n) L/D=6,Vr=6

31  

Fig. 20 Contours of the real part of the wavelet transform of the vibration displacement in the time-

frequency plane for two cylinders in the side-by-side arrangement

For L/D=1.5 and 2, the response amplitude at reduced velocities higher than 15 are found to be

highly irregular. Both vibration amplitude and vibrations change with time. Morlet wavelet analysis,

which can identify the evolution of both amplitude and frequency, is used to analyse the response of

the cylinder system. In the time domain the complex Morlet wavelet is expressed as

2/2/4/1 2200 )()( tti eeet , where ω0 is the nondimensional frequency, which is taken to be 6

to satisfy the admissibility condition [46, 47]. The Fourier transform of Morlet wavelet is

2/)(4/1 202)(ˆ e and the continuous wavelet transform (CWT) of a signal X(t) is defined as

32  

d]/)[()(/1),( atXaatWx where a is the time scale. The local amplitude spectrum is

calculated by SatWatA xx /|),(|2),( , where S is the peak amplitude of the Fourier transform of the

sampled wavelet [48]. Fig. 20 shows the contours of the real part of the wavelet transform of the

vibration displacement in the time-frequency plane for two cylinders in the side-by-side

arrangement. The variation of the frequency with time can be clearly identified in Fig. 20. Take Fig.

20 (a) as an example, the amplitude of displacement at frequency of f/fn=2 become strong and weak

intermittently. The component with a frequency of about 0.5-0.7 during 5<tfn<10 and 32<tfn<37 and

82<tfn<87 and is very weak in other time. Based on the wavelet analysis, it is found that vibrations

corresponding to Fig. 20 are the combination of VIV and galloping. Galloping is a classical

instability mechanism where a small transverse body motion creates an aerodynamic force that

increases the motion [49]. It is characterized by low-frequency oscillations that increase in amplitude

unbounded with fluid velocity [50]. Galloping is driven by a time-averaged fluid force which

develops in phase with the structural velocity and has a frequency many times lower than the vortex

shedding frequency [51]. In Fig. 20 (a) and (b), the highest frequency varies between 1.5 and 2.5.

The Strouhal numbers for two side-by-side stationary cylinders at L/D=1.5 and 2 are St=1.66 and

0.172, respectively. If the nondimensional vibration frequency follows the Strouhal law, it should be

trn/ SVff . The highest frequencies in Fig. 20 (a), (b) and (c) are believed to be the frequencies

lead by vortex shedding. They are slightly less than the frequencies obtained from the Strouhal law

due to the effects of the vibration. The high frequency component can be also clearly seen in Fig. 19

(e), (f), (i) and (j). In each of the wavelet spectrum in Fig. 20, the low frequency component which is

between 0.5 and 1 are also observed. This frequency component, which is much lower than the

vortex shedding frequency, is believed to be the results of galloping. The galloping is explained by

Fig. 27 and associated discussion about Fig. 27 later on. The variation of the vibration frequency

with time can be clearly seen in Fig. 20, resulting in very irregular vibrations. It appears that the high

frequency component becomes weak and the low frequency component becomes strong with an

33  

increase in reduced velocity. This means that the galloping becomes strong and the VIV becomes

weak as reduced velocity increases. The galloping amplitude increases and the VIV amplitude

decreases can also be seen in the time histories in Fig. 19. Take L/D=2 as an example, the response is

dominated by the high frequency component at Vr=13.5 and the low frequency component at Vr=30

(Fig. 19 (h) and (j)). In Fig. 19 (d) the high amplitude components has been extremely weak.

 

Fig. 21 Variation of the response amplitude with the reduced velocity for the side-by-side

arrangement 

 

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

A

Side-by-side, L/D=1.5

Side-by-side, L/D=2

Single cylinder

(a) Amplitude for L/D=1.5 and 2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A

Vr

Side-by-side, L/D=4

Side-by-side, L/D=6

Single cylinder

(b) Amplitude for L/D=4 and 6

34  

Fig. 21 shows the variation of the response amplitude with the reduced velocity for the side-by-

side arrangement. The maximum calculated reduced velocity for L/D=1.5 and 2 are 30 in order to

identify the galloping regime. Fig. 22 shows the FFT spectra of the displacement and the lift

coefficient for two cylinders in the side-by-side arrangement. Fig. 23 shows the variation of the

response frequency with the reduced velocity for the side-by-side arrangement. The frequencies

shown in Fig. 23 are the peak frequencies of the amplitude spectra. For L/D=1.5 and 2, because the

vibrations at Vr>15 are so irregular that no distinct peak frequencies can be identified in the FFT

spectra, only the frequencies at 15r V are shown in Fig. 23 (a). The lock-in regimes for L/D=1.5

and 2 are found to be about 155.3 r V , which are much wider than that of a single cylinder. The

response amplitude for L/D=1.5 in the lock-in regime increases very slowly with the increasing

reduced velocity and peaks at about Vr=8.5. The maximum amplitude in the lock-in regime for

L/D=0.5 is almost 1, which is about 70% higher than that of a single cylinder. For L/D=1.5 and 2, the

response amplitude increases almost monotonically with the increasing reduced velocity as Vr>15

until the maximum simulated reduced velocity of 30. In the study of galloping of a square cylinder

Nemes et al. [50] found that the response frequency does not change with the reduced velocity and is

close to the natural frequency of the cylinder. Although the vibration frequency at Vr>15 cannot be

identified due to the strong irregularity of the vibration, it can be seen in the wavelet spectra in Fig.

20 that the nondimensional galloping frequency is generally less than 1 and much smaller than the

vortex shedding frequency.

35  

Fig. 22 FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side

arrangement

For two cylinders in the side-by-side arrangement at L/D=4 and 6, the response amplitude and

frequency have been very similar to that of a single cylinder. The lock-in regime becomes closer to

that of a single cylinder as the gap ratio L/D is increased from 4 to 6. Different from that of a single

cylinder, the response amplitude in the range of 3r V is exactly zero for L/D=4 and 6. The

amplitude is exactly zero because the flow is exactly symmetric as discussed later on. Outside the

lock regime the response frequency increases linearly with the reduced velocity, indicating that it

follows the Strouhal law. Based on the response amplitude and frequency it can be seen that for all

the gap ratios, the lower boundaries of the lock-in regime are the same and they are 3.5.

0

0.5

1

1.5

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(a) L/D=1.5, Vr=3.5

0

0.5

1

1.5

0

0.5

1

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(b) L/D=1.5, Vr=5

0

0.5

1

1.5

0

0.5

1

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(c) L/D=1.5, Vr=8.5

0

0.2

0.4

0.6

0

0.5

1

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(d) L/D=1.5, Vr=12

0

0.1

0.2

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(e) L/D=1.5, Vr=18

0

1

2

3

0

0.2

0.4

0.6

0.8

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(f) L/D=2, Vr=4

0

0.1

0.2

0.3

0

0.2

0.4

0.6

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(g) L/D=2, Vr=10

0

0.1

0.2

0

0.1

0 1 2 3 4

AC

L

Ay/

D

f/fn

Y

CL

(h) L/D=2, Vr=15

0

0.1

0.2

0

0.1

0.2

0 1 2 3 4

AC

L

Ay/

D

f/fn

Y

CL

(a) L/D=2, Vr=25

36  

Fig. 23 Variation of the response frequency with the reduced velocity for the side-by-side

arrangement

 

Fig. 24 shows the variation of the RMS lift coefficient on the two cylinders in the tandem

arrangement with the reduced velocity. The RMS lift coefficient starts to increases with the

increasing reduced velocity at about Vr=3.5 for all the gap ratios, which is the lower boundary of the

lock-in regime. For a single cylinder, after the RMS lift coefficient peaks at Vr=3.5, it reduces very

quickly to a very small value at Vr=5.5. It can be seen that both the RMS lift coefficient for each

individual cylinder and the total RMS lift coefficient are significantly greater than that of a single

cylinder in the reduced velocity range of 155.5 r V , resulting in much wider lock-in regime than

that of a single cylinder. As Vr>15, the RMS lift coefficient is almost independent on the reduced

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Side-by-side, L/D=1.5

Side-by-side, L/D=2

Single cylinder

(a) Frequency for L/D=1.5 and 2

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Vr

Side-by-side, L/D=4

Side-by-side, L/D=6

Single cylinder

(b) Frequency for L/D=4 and 6

37  

velocity for L/D=1.5 and 2. The RMS lift coefficient of the total force for L/D=1.5 as 155.3 r V is

slightly greater than that for L/D=2. For L/D=4 and 6, the variations of the RMS lift coefficient with

the reduced velocity is very similar to that of a single cylinder. The RMS lift coefficient of the total

force as 3r V is zero because the vortex shedding flow is exactly symmetric as shown in Fig. 30.

Fig. 24 RMS lift coefficient on the two cylinders in the side-by-side arrangement 

 

Fig. 25 shows the variation of the phase ψ between the response displacement and the total lift

force with the reduced velocity for two cylinders in the side-by-side arrangement. Similar to that of a

single cylinder, the phase ψ jumps from 0° to 180° at a reduced velocity for each gap ratio. The

phase changes from 0° to 180° at Vr = 8.5 and 10, respectively for L/D=1.5 and 2. At L/D=4 and 6

the phase changes happens at Vr=5.5. The phase ψ for a gap ratio with narrower lock-in regime

changes from 0° to 180° at smaller reduced velocity than that for a gap ratio with wider lock-in

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CL

,rm

s

Side-by-side L/D=1.5

Side-by-side L/D=2

Single cylinder

(a) CL,rms of Cylinder 1 for L/D=1.5 and 2

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15C

L,r

ms

Side-by-side L/D=4

Side-by-side L/D=6

Single cylinder

(b) CL,rms of Cylinder 1 for L/D=4 and 6

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CL

,rm

s

Side-by-side L/D=1.5

Side-by-side L/D=2

Single cylinder

(c) CL,rms of Cylinder 2 for L/D=1.5 and 2

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Side-by-side L/D=4

Side-by-side L/D=6

Single cylinder

(d) CL,rms of Cylinder 2 for L/D=4 and 6

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CL

,rm

s

Vr

Side-by-side L/D=1.5

Side-by-side L/D=2

Single cylinder

(e) CL,rms of total force for L/D=1.5 and 2

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Vr

Side-by-side L/D=4

Side-by-side L/D=6

Single cylinder

(f) CL,rms of total force for L/D=4 and 6

38  

regime. The variations of phase with the reduced velocity at L/D=4 and 6 are very similar to each

other due to the weak effects of the gap on the flow and VIV.

Fig. 25 Variation of the phase between the response displacement and the total lift force with the

reduced velocity for two cylinders in the side-by-side arrangement 

Fig. 26 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5

and Vr=8.5, where the response amplitude in the cross-flow direction peaks. In Fig. 26 (a) and (b),

the positive vortex from the bottom of the top cylinder is very weak and the two negative vortices

from the tops of the two cylinders combine. The combined negative vortex starts to be shed from the

cylinder in Fig. 26 (c), followed by the combination of the two positive vortices from the bottoms of

the two cylinders. The vortex shedding from the bottom of the cylinders when the cylinders are at

their lowest position makes a negative lift force on the cylinder system. This makes the lift force in-

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Side-by-side, L/D=1.5

Side-by-side, L/D=2

(a) Phase between Y and CL for L/D=1.5 and 2

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Vr

Side-by-side, L/D=4

Side-by-side, L/D=6

(b) Phase between Y and CL for L/D=4 and 6

39  

phase with the response displacement. In each vibration cycle, two main vortices are shed from the

cylinder and the vortex street in the wake of the two cylinders is very similar to that of a single

cylinder. The two cylinders behave like a single body. The lift coefficients of the two cylinders

shown at the bottom of Fig. 26 are more or less in anti-phase with each other, leading to a total lift

coefficient with amplitude smaller than that of either of the cylinders. The lift coefficient of the total

force is in phase with the vibration displacement. In Fig. 26 (d), very small portion of the positive

vortex generated from the bottom of the top cylinder is cut off from its main part. This vortex is too

weak to have effect on the VIV.

Fig. 26 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5 and Vr=8.5

-4

-3

-2

-1

0

1

2

3

4

-1.5

-1

-0.5

0

0.5

1

1.5

248 248.2 248.4 248.6 248.8 249 249.2 249.4 249.6 249.8 250

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a)(b) (c) (d) (e) (f)

40  

Fig. 27 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5

and Vr=25, where the galloping occurs. The vortices from the gap between the cylinders are stronger

than those in Fig. 26, making it harder for the two vortices of the same direction from the two

cylinders to combine with each other. It can be seen in Fig. 27 (a) and (b) that after the negativee

vortex is shed from the top surface of the top cylinder, it is connected with the negative vortex from

the top surface of the bottom cylinder, which has been separated from the cylinder surface. These

two negative vortices are separated from each other in Fig. 27 (c), due to the combination effects of

the two positive vortices from the bottoms of the cylinders. The vortex shedding flow in one

vibration period is found different from that in another vibration period. It is found that whenever the

galloping occurs (Fig. 27 (a) – (g)), the vortices from the gap between the two cylinders are strong.

As the response amplitude is increased from its lowest position in Fig. 27 (c) to its highest position in

Fig. 27 (g), the lift coefficients on both cylinders oscillate about three periods. Apart from the vortex

shedding, it is found that the change of the direction of the jet flow from the gap between the

cylinders also contributes the oscillation of the lift coefficient. For example, the jet flow between the

gap in Fig. 27 (c) biases upwards and that in Fig. 27 (d) biases downwards. Although there is not

obvious vortex shedding, the lift coefficients of both cylinder oscillates almost one period between

Fig. 27 (c) and (d). The switching of the gap flow from one side to the other is known as flip-

flopping flow [15, 16, 42]. The vortex shedding flow in Fig. 27 (h), (i) and (j), which is within the

regular VIV period between tfn=53.7 to 54.7, is very similar to that in Fig. 26. After tfn=55, galloping

occurs again with the response displacement continuously increasing for a long period of time.

41  

Fig. 27 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5 and Vr=25

 

-2

-1

0

1

2

-1

-0.5

0

0.5

1

52 52.5 53 53.5 54 54.5 55 55.5 56

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a)(b)(c)(d)(e)(f) (g) (j)(h) (i)

42  

Fig. 28 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=2

and Vr=4.5, where the response amplitude peaks. Compared with those for L/D=1.5, vortex shedding

flows from the two cylinders are more independent on each other. The negative vortices from the

tops of the two cylinders are shed from the cylinders separately without combination as shown in

Fig. 28 (a), (b) and (c). The two positive vortices from the bottoms of the cylinders are also shed

from the cylinders separately. They combined after they are shed from the cylinders. The vortices

interact with each other after they are shed from the cylinders, resulting very irregular vortex street.

Fig. 28 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=2 and Vr=4.5

 

-3

-2

-1

0

1

2

3

-1

-0.5

0

0.5

1

319 319.2 319.4 319.6 319.8 320 320.2 320.4 320.6 320.8 321

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d) (e) (f)

43  

Fig. 29 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=4

and Vr=6. The interaction between the two cylinders has been very weak at L/D=4 and the vortex

shedding flows from the two cylinders are in phase with each other. In each vibration period, two

vortices (a pair) are shed from each cylinder. The interaction between the two vortices of the same

direction from the two cylinder may occurs, but in the far downstream area of the cylinders. The

vortex shedding from each cylinder is very similar to that of a single cylinder and the vibration

frequency is the same as the vortex shedding frequency. The vortex shedding from each cylinder is

in 2S mode.

Fig. 29 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and Vr=6

Fig. 30 shows the vorticity contours for two cylinders in a side-by-side arrangement at L/D=6,

Vr=3 and L/D=4, Vr=3, where the response amplitude is zero. The response amplitude is zero because

the vortex shedding from one cylinder is wholly in anti-phase with that from another cylinder. The

anti-phase between the two vortex streets leads zero lift coefficient and in turn no vibration of the

-1

0

1

-1

-0.5

0

0.5

1

481.5 482 482.5 483 483.5 484 484.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

44  

cylinders. In order to testify if the anti-phase between the two vortex streets is intrinsic, separate

simulations are conducted for Vr=0.5 to 3 at L/D=4 and 6 with an initial displacement of the cylinder

in the cross-flow direction. Fig. 30 (c) is an example of the time history of the vibration displacement

of the cylinders for L/D=6 and Vr=3. It can be seen that the initial displacement of the cylinders

decreases gradually with time and finally become zero. Fig. 30 (c) demonstrates that the flow pattern

in the wake of the two cylinders is intrinsically symmetric. Even an asymmetric initial condition will

leads to symmetric flow pattern.

Fig. 30 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=6, Vr=3 and

L/D=4, Vr=3

6. Conclusions

Flow-induced vibration of two cylinders in tandem and side-by-side arrangements at a low

Reynolds number of 150 are studied numerically by solving the 2D Navier-Stokes equations using

FEM. The numerical model is carefully validated against the published data of flow past two fixed

cylinders, VIV of a single cylinder and VIV of one elastically mounted cylinder in the wake of the

fixed cylinder. The conclusions are summarized as follows.

-1

0

1

0 50 100 150 200 250

Y/D

tfn

(c) Time histories for L/D=6, Vr=3. Initial disturbation is introduced.

45  

6.1 Tandem arrangement

When the two cylinders are in the tandem arrangement, the critical gap for vortex shedding from

the upstream cylinder is significantly less than that for two stationary cylinders. If the two cylinders

are stationary, the vortex shedding from the upstream cylinder does not occur until the gap ratio L/D

exceeds 3.5 or 4. However, if the two cylinders are elastically mounted, the vortex shedding from the

upstream cylinder happens at L/D=2 in the lock-in regime.

In the tandem arrangement, the lock-in regime of the reduced velocity is narrower than that of a

single cylinder for L/D=1.5 and 2 and wider than later for L/D=4 and 6. The response amplitude at

L/D=1.5 is much smaller than that of a single cylinder. For L/D=4 and 6, the maximum response

amplitude in the lock-in regime is much greater than that of a single cylinder. For L/D=6 and in the

range of 5.139 r V , the lift force has two frequency components and because secondary lift

component locks on the natural frequency, the response amplitude in the lock-in regime is small. The

vortex shedding from the upstream cylinder has significant effects on the response. The vortices that

are shed from the upstream cylinder may scratch the top and the bottom surface or impinge the front

surface of the downstream cylinder, depending on the reduced velocity.

6.2 Side-by-side arrangement

The most interesting phenomenon found in the side-by-side arrangement is the combination of

VIV and galloping at L/D=1.5 and 2. For L/D=1.5 and 2, the response is dominated by VIV as Vr<15

and by galloping as Vr>15. At reduced velocities close to 15, the response is a combination of VIV

and galloping. The Wavelet transform is used to identify that the vibration frequencies at reduced

velocities greater than 15 include the vortex shedding frequency and the galloping frequency. With

an increase in the reduced velocity, the amplitude corresponding to VIV becomes weak and that

46  

corresponding to the galloping becomes strong. Due to galloping the response amplitude increases

continuously with the increasing reduced velocity as Vr>15.

The response of the two cylinder at large L/D=4 and 6 is very similar to that of a single cylinder

due to the weak interaction between the two vortex streets. The vortex shedding flows at these two

L/D are in-phase with each other in the lock-in regime. They are in anti-phase with each other as

3rV even with an asymmetric initial condition, resulting in absolutely no vibration.

The vortex shedding from the two cylinders for L/D=1.5 and 2 is very similar to that of a single

cylinder in the VIV regime. During the vortex shedding, the two negative vortices from the cylinders

combine and the two positive vortices combine, leaving two rows of vortices in the wake of the

cylinders. When the response is in the galloping regime, apart from the vortex shedding, the change

in the jet flow direction also leads to the oscillatory of the lift force. The galloping frequency is much

smaller than the frequency of the lift force.

References

1. C.H.K.Williamson and A. Roshko, “Vortex formation in the wake of an oscillating cylinder,”

Journal of Fluids and Structures 2, 355–381 (1988).

2. T. Sarpkaya, “A critical review of the intrinsic nature of vortex-induced vibrations,” Journal of

Fluids and Structures 19 (4), 389-447 (2004).

3. C.H.K Williamson and R. Govardhan, “Vortex-induced vibrations,” Annual Review of Fluid

Mechanics 36, 413–455 (2004).

4. A. Khalak, and C.H.K Williamson, “Motions, force and mode transitions in vortex-induced

vibrations at low mass-damping,” Journal of Fluids and Structures 13, 813-851 (1999).

5. N. Jauvtis and C.H.K Williamson, “The effect of two degrees of freedom on vortex-induced

vibration at low mass and damping,” Journal of Fluid Mechanics 509, 23-62 (2004).

6. R. Govardhan and C.H.K Williamson, “Modes of vortex formation and frequency response of a

freely vibrating cylinder,” Journal of Fluid Mechanics 420, 85-130 (2000).

7. S.P. Singh and S. Mittal, S., 2005, “Vortex-induced oscillations at low Reynolds numbers:

Hysteresis and vortex-shedding modes,” Journal of Fluids and Structures 20, 1085–1104 (2005).

47  

8. M. Zhao, L. Cheng, and T. Zhou, T., “Numerical simulation of vortex-induced vibration of a

square cylinder at a low Reynolds number,” Physics of Fluids 25, 023603 (2013).

9. C.C.Feng, C. C., “The measurements of vortex-induced effects on flow past a stationary and

oscillation and galloping,” Master’s thesis, University BC, Vancouver, Canada (1968).

10. P.W. Bearman, “Vortex shedding from oscillating bluff_bodies,” Annual Review of Fluid

Mechanics 16, 195-222 (1984).

11. M.M. Zdravkovich, “Review of flow interference between two circular cylinders in various

arrangements,” ASME Journal of Fluids Engineering 99, 618-633 (1977).

12. M.M. Zdravkovich, “Flow induced oscillations of two interfering circular cylinders,” Journal of

Sound and Vibration 101, 511-521 (1985).

13. M.M. Zdravkovich, “The effects of interference between circular cylinders in cross flow,”

Journal of Fluids and Structures 1, pp. 239-261 (1987).

14. C.H.K. Williamson, “Evolution of a single wake behind a pair of bluff bodies,” Journal of Fluid

Mechanics 159, 1-18 (1985).

15. P.W. Bearman and A.J. Wadcock, “The interaction between a pair of circular cylinders normal

to a stream,” Journal of Fluid Mechanics 61, 499-511 (1973).

16. H.J. Kim and P.A. Durbin, “Investigation of the flow between a pair of cylinders in the flopping

regime,” Journal of Fluid Mechanics, 196, 431-448 (1988).

17. M.M. Alam, M. Moriya, and H. Sakamoto, “Aerodynamic characteristics of two side-by-side

circular cylinders and application of wavelet analysis on the switching phenomenon,” Journal of

Fluids and Structures 18, 325–346 (2003).

18. J.R. Meneghini, F. Saltara, C.L.R. Siqueira, and J.J.A Ferrari, “Numerical simulation of flow

interference between two circular cylinders in tandem and side-by-side arrangements,” Journal

of Fluids and Structures 15 (2), 327-350 (2001).

19. J. Mizushima and N. Suehiro, “Instability and transition of flow past two tandem circular

cylinders,” Physics of Fluids 17 (10), 104107 (2005).

20. Y. Tasaka, S. Kon, L. Schouveiler, and P.L. Gal, “Hysteretic mode exchange in the wake of two

circular cylinders in tandem,” Physics of Fluids 18, 084104 (2006).

21. M.M. Zdravkovich, “Review of interference-induced oscillations in flow past two parallel

circular cylinders in various arrangements,” J. Wind Engineering and Industrial Aerodynamics

28, 183–200 (1988)

22. E. Fontaine, J. Morel, Y. Scolan, and T. Rippol, “Riser interference and VIV amplification in

tandem configuration,” International Journal of Offshore and Polar Engineering 16 (1), 33–40

(2006).

48  

23. H.P. Ruscheweyh, “Aeroelastic interference effects between slender structures,” Journal of

Wind Engineering and Industrial Aerodynamics. 14 (1–3), 129-140 (1983).

24. D. Brika, and A. Laneville, “The flow interaction between a stationary cylinder and a

downstream flexible cylinder,” Journal of Fluids and Structures 13 (5), 579-606 (1999).

25. A. Bokaian and F. Geoola, “Wake-induced galloping of two interfering circular cylinders,”

Journal of Fluid Mechanics 146, 383-415 (1984).

26. S. Mittal and V. Kumar, “Flow-induced oscillations of two cylinders in tandem and staggered

arrangements,” Journal of Fluids and Structures 15 (5), 717-736 (2001).

27. S. Mittal and V. Kumar, V., “Vortex induced vibrations of a pair of cylinders at Reynolds

number 1000,” International Journal of Computational Fluid Dynamics 18, 601-614 (2004).

28. W. Jester and Y. Kallinderis, “Numerical study of incompressible flow about transversely

oscillating cylinder pairs,” Journal of Offshore Mechanics and Arctic Engineering 126 (4), 310-

317 (2004).

29. Z. Zang, F. Gao, and J.S. Cui, 2013, “Physical modeling and swirling strength analysis of vortex

shedding from near-bed piggyback pipelines,” Applied Ocean Research 40, 50-59 (2013).

30. M. Zhao and G. Yan, “Numerical simulation of vortex-induced vibration of two circular

cylinders of different diameters at low Reynolds number,” Physics of Fluids 25, 083601 (2013).

31. F.J. Huera-Huarte, and M. Gharib, M., “Flow-induced vibrations of a side-by-side arrangement

of two flexible circular cylinders,” Journal of Fluids and Structures 27, 354–366 (2011).

32. X.Q. Wang, R.M.C. So, W.C. Xie, and J. Zhu, J., “ Free-stream turbulence effects on vortex-

induced vibration of two side-by-side elastic cylinders,” Journal of Fluids and Structures 24,

664-679 (2008).

33. C.H.K. Williamson, “The existence of two stages in the transition to three-dimensionality of a

cylinder wake,” Physics of Fluids 31, 3165–3168 (1988).

34. C.H.K. Williamson, “Oblique and parallel modes of vortex shedding in the wake of a circular

cylinder at low Reynolds numbers,” Journal of Fluid Mechanics 206, 579-627 (1989).

35. I. Borazjani and F. Sotiropoulos, “Vortex-induced vibrations of two cylinders in tandem

arrangement in the proximity-wake interference region,” Journal of Fluid Mechanics 621, 321-

364 (2009).

36. M. Zhao and L. Cheng, “Numerical simulation of two-degree-of-freedom vortex-induced

vibration of a circular cylinder close to a plane boundary,” Journal of Fluids and Structures 27,

1097–1110 (2011).

37. M. Zhao and L. Cheng, “Numerical investigation of local scour below a vibrating pipeline under

steady currents,” Coastal Engineering 57, 397–406 (2010).

49  

38. M. Zhao, F. Tong, and L. Cheng, “Numerical simulation of two-degree-of-freedom vortex-

induced vibration of a circular cylinder between two lateral plane walls in steady currents,”

Journal of Fluids Engineering, 134, 104501 (2012).

39. Y. Bao, D. Zhou, and J. Tu, “Flow interference between a stationary cylinder and an elastically

mounted cylinder arranged in proximity,” Journal of Fluids and Structures 27, 1425–1446

(2011).

40. H.T. Ahn and Y. Kallinderis, “Strongly coupled flow/structure interactions with a geometrically

conservative ALE scheme on general hybrid meshes,” Journal of Computational Physics 219,

671–696 (2006).

41. G.R.S. Assi, J.R. Meneghini, J.A.P. Aranhaa, P.W. Bearman, and E. Casaprima, “Experimental

investigation of flow-induced vibration interference between two circular cylinders,” Journal of

Fluids and Structures 22, 819–827 (2006).

42. M.M. Zdravkovich and D.L. Pridden, “Interference between two circular cylinders, series of

unexpected discontinuities,” Journal of Industrial Aerodynamics, 2, 255-270 (1977).

43. T.K., Prasanth, S. Sehara, S.P. Singh, R. Kumar, and S. Mittal, “ Effect of blockage on vortex-

induced vibrations at low Reynolds numbers,” Journal of Fluids and Structures 22, 865-876

(2006).

44. B.M. Sumer and J. Fredsøe, Hydrodynamics around Cylindrical Structures. World Scientific

(1997).

45. S. Mittal and V. Kumar, “Finite element study of vortex-induced cross-flow and in-line

oscillations of a circular cylinder at low Reynolds numbers,” International Journal for Numerical

Methods in Fluids 31, 1087-1120 (1999).

46. M. Farge, “Wavelet transforms and their applications to turbulence,” Annual Review of Fluid

Mechanics 24, 395-457 (1992).

47. R.N., Leao and J.A. Burne, J.A., “Continuous wavelet transform in the evaluation of stretch

reflex response from surface EMG,” Journal of Neuroscience Methods 133, 115-125 (2004).

48. B. Nener, C. Ridsdill-Smith, and T.A. Zeisse, “Wavelet analysis of flow altitude infrared

transmission in the coastal environment,” Infrared Physics & Technology 40, 399-409 (1999).

49. P.W. Bearman, I.S Gartshore, D.J. Maull, and G.V. Parkinson, “Experiments on flow-induced

vibration of a square-section cylinder,” Journal of Fluids and Structures 1, 19-34 (1987).

50. A. Nemes, J. Zhao, D.L. Jacono, and J. Sheridan, “The interaction between flow-induced

vibration mechanisms of a square cylinder with varying angles of attack,” Journal of Fluid

Mechanics 710, 102-130 (2012).

50  

51. I. Robertson, L. Li, S.J. Sherwin, and P.W Bearman, P.W., “A numerical study of rotational and

transverse galloping rectangular bodies,” Journal of Fluids and Structures 17, 681–699 (2003).

51  

List of Figure captions

Fig. 1 Sketch of flow past two cylinders in (a) tandem and (b) side-by-side arrangements

Fig. 2 Computational mesh near the two cylinders

Fig. 3 Variations of the response amplitude and the force coefficients with the reduced velocity for a

single cylinder at Re=150, Mred=2 and ζ=0.

Fig. 4 The response amplitude and the root mean square lift coefficient of an elastically mounted

cylinder in the wake of stationary cylinder with L/D=4, Mred=2 and ζ=0

Fig. 5 Variations of the mean drag and root mean square lift coefficients with L/D for two stationary

cylinders in the tandem arrangement

Fig. 6 Time histories of the vibration displacement for the tandem arrangement

Fig. 7 Amplitude spectra of the vibration based on the FFT analysis

Fig. 8 Variation of the response amplitude with the reduced velocity for the tandem arrangement

Fig. 9 Variation of the response frequency with the reduced velocity for the tandem arrangement

Fig. 10 Variation of the RMS lift coefficient with the reduced velocity for two cylinders in the

tandem arrangement

Fig. 11 Variation of the phase between the response displacement and the total lift force with the

reduced velocity for tandem arrangement

Fig. 12 Vorticity contours around two cylinders in a tandem arrangement at L/D=1.5 and Vr=4.5

Fig. 13 Vorticity contours for two cylinders in a tandem arrangement at L/D=2 and Vr=5

Fig. 14 Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and Vr=6

Fig. 15 Vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the

lowest location for L/D=4 and Vr=4, 4.5, 5 and 7

Fig. 16 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=4

Fig. 17 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=8

Fig. 18 Variations of mean drag and root mean square lift coefficients with L/D for two cylinders in

the side-by-side arrangement at Re=150

Fig. 19 Time histories of the cylinder displacement for the side-by-side arrangement

Fig. 20 Contours of the real part of the wavelet transform of the vibration displacement in the time-

frequency plane for two cylinders in the side-by-side arrangement

Fig. 21 Variation of the response amplitude with the reduced velocity for the side-by-side

arrangement

Fig. 22 FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side

arrangement

52  

Fig. 23 Variation of the response frequency with the reduced velocity for the side-by-side

arrangement

Fig. 24 RMS lift coefficient on the two cylinders in the side-by-side arrangement

Fig. 25 Variation of the phase between the response displacement and the total lift force with the

reduced velocity for two cylinders in the side-by-side arrangement

Fig. 26 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5 and Vr=8.5

Fig. 27 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5 and Vr=25

Fig. 28 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=2 and Vr=4.5

Fig. 29 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and Vr=6

Fig. 30 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=6, Vr=3 and

L/D=4, Vr=3

 

 

Table 1 Comparison of the numerical results of flow past two stationary cylinders in tandem Reference L/D 1DC ACL1 St1 2DC ACL2 St2

Borazjani and Sotiropoulos (2009)

1.5 1.112 0.024 0.174 -0.216 0.050 0.174

Meneghini et al. (2001) 1.060 0.03 0.167 -0.180 0.060 0.167 Present 1.086 0.025 0.166 -0.198 0.063 0.166

Meneghini et al. (2001) 2

1.030 0.070 0.130 -0.170 0.149 0.130 Present 1.046 0.030 0.131 -0.208 0.106 0.130

Borazjani and Sotiropoulos (2009)

4 1.290 0.745 0.185 0.600 1.900 0.185

Meneghini et al. (2001) 1.180 0.75 0.174 0.380 1.500 0.174 Present 1.269 0.821 0.172 0.366 1.607 0.172

Fig. 1 Sketch of flow past two cylinders in (a) tandem and (b) side-by-side arrangements

x

y

O

L

DDFlow

(a)

Cylinder 1 Cylinder 2

Cylinder 1

x

y

OL

D

D

Flow

(b)Cylinder 2

Fig. 2 Computational mesh near the cylinders

Fig. 3Variations of the response amplitude and the force coefficients with the reduced velocity for a single cylinder at Re=150, Mred=2 and ζ=0.

0

0.2

0.4

0.6

0.8

1

2 3 4 5 6 7 8 9

Ay/

D

Vr

Ahn and Kallinderis [40]Borazjani and Sotiropoulos [35]Bao et al. [39]Present

(a) Response amplitude

0

0.5

1

1.5

2

2.5

3

2 3 4 5 6 7 8 9

For

ce c

oeff

icie

nt

Vr

Mean CD, Bao et al. [39]Mean CD, PresentRMS CL, Bao et al. [39]RMS CL, PresentRMS CD, Bao et al. [39]RMS CD, Present

(b) Force coefficient

Fig. 4 The response amplitude and the root mean square lift coefficient of an elastically mounted cylinder in the wake of stationary cylinder with L/D=4, Mred=2 and ζ=0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

3 4 5 6 7 8 9 10 11 12

Ay/

D

Vr

Bao et al. [39]PresentAssi et al. [41]

(a) Response amplitude of the downstreamcylinder

m*≈2, m*ζ≈3, Re=3000 - 13000

0

0.5

1

1.5

3 4 5 6 7 8 9 10 11 12

CL

,rm

s

Vr

Bao et al. [39]

Present

(b) Root mean square of the lift coefficientof the downstream cylinder

Fig. 5 Variations of the mean drag and root mean square lift coefficients with L/D for two stationary cylinders in the tandem arrangement

-0.5

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6

CD

L/D

Upstream cylinderDownstream cylinderSingle cylinderExp., downstream cylinder, Re=31000

(a) Mean Drag coefficient

[42]

0

0.5

1

1.5

2

1 2 3 4 5 6

CL

,rm

s

L/D

Upstream cylinderDownstream cylinderSingle cylinder

(b) RMS lift coefficient

Fig. 6 Time histories of the cylinder displacement for the tandem arrangement

-0.4

-0.2

0

0.2

0.4

250 260 270 280 290 300 310 320 330 340 350

Y/D

(a) L/D=1.5,Vr=4.5

-0.4

-0.2

0

0.2

0.4

250 260 270 280 290 300 310 320 330 340 350

Y/D

(b) L/D=1.5,Vr=5

-1

-0.5

0

0.5

1

250 260 270 280 290 300 310 320 330 340 350

Y/D

(c) L/D=2,Vr=4

-1

-0.5

0

0.5

1

200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

Y/D

(d) L/D=2,Vr=5

-1

-0.5

0

0.5

1

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

Y/D

(e) L/D=2,Vr=6

-1

-0.5

0

0.5

1

200 210 220 230 240 250 260 270 280 290 300

Y/D

(f) L/D=2,Vr=6.5

-1

-0.5

0

0.5

1

80 90 100 110 120 130 140 150 160 170 180

Y/D

(g) L/D=4,Vr=6

-1

-0.5

0

0.5

1

80 90 100 110 120 130 140 150 160 170 180Y/

D

(h) L/D=4,Vr=7.5

-1

-0.5

0

0.5

1

80 90 100 110 120 130 140 150 160 170 180

Y/D

(i) L/D=6,Vr=6

-1

-0.5

0

0.5

1

40 50 60 70 80 90 100 110 120 130 140

Y/D

(j) L/D=6,Vr=7.5

-0.4

-0.2

0

0.2

0.4

60 70 80 90 100 110 120

Y/D

tfn

(k) L/D=6,Vr=9

-0.4

-0.2

0

0.2

0.4

50 60 70 80 90 100 110

Y/D

tfn

(l) L/D=6,Vr=10

Fig. 7 Amplitude spectra of the vibration based on the FFT analysis

0

0.2

0.4

0.6

0.8

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(a) L/D=1.5, Vr=4.5

0

0.2

0.4

0.6

0.8

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(b) L/D=1.5, Vr=5

0

0.5

1

1.5

0

0.1

0.2

0.3

0.4

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(c) L/D=2, Vr=5

0

0.1

0.2

0.3

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(d) L/D=2, Vr=6

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(e) L/D=6, Vr=9

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(f) L/D=6, Vr=10

Fig. 8 Variation of the response amplitude with the reduced velocity for the tandem arrangement

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ay/

D

Tandem, L/D=1.5

Tandem, L/D=2

Single cylinder

(a) Amplitude for L/D=1.5 and 2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ay/

D

Vr

Tandem, L/D=4

Tandem, L/D=6

Single cylinder

(b) Amplitude for L/D=4 and 6

Fig. 9 Variation of the response frequency with the reduced velocity for the tandem arrangement

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Tandem, L/D=1.5

Tandem, L/D=2

Single cylinder

(a) Frequency for L/D=1.5 and 2

f/fn=1

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Vr

Tandem, L/D=4

Tandem, L/D=6

Single cylinder

(b) Frequency for L/D=4 and 6

f/fn=1

Fig. 10 Variation of the RMS lift coefficient with the reduced velocity for two cylinders in the tandem arrangement

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Tandem, L/D=1.5

Tandem, L/D=2

Tandem, L/D=4

Tandem, L/D=6

(a) CL,rms of the upstream cylinder

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Tandem, L/D=1.5

Tandem, L/D=2

Tandem, L/D=4

Tandem, L/D=6

(b) CL,rms of the downstream cylinder

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Vr

Tandem, L/D=1.5

Tandem, L/D=2

Tandem, L/D=4

Tandem, L/D=6

(c) CL,rms of the total force

Fig. 11 Variation of the phase between the response displacement and the total lift force with the reduced velocity for tandem arrangement

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Tandem, L/D=1.5

Tandem, L/D=2

(c) Phase between Y and CL for L/D=1.5 and 2

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Vr

Tandem, L/D=4

Tandem, L/D=6

(c) Phase between Y and CL for L/D=4 and 6

Fig. 12 Vorticity contours for two cylinders in a tandem arrangement at L/D=1.5 and Vr=4.5

-1.5

-1

-0.5

0

0.5

1

1.5

-0.4

-0.2

0

0.2

0.4

528 528.5 529 529.5 530 530.5 531

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

Fig. 13 Vorticity contours for two cylinders in a tandem arrangement at L/D=2 and Vr=5

-4

-3

-2

-1

0

1

2

3

4

-1

-0.5

0

0.5

1

360 361 362 363 364 365 366 367 368 369C

L

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a) (b) (c) (d) (e) (f)

Fig. 14 Vorticity contours for two cylinders in a tandem arrangement at L/D=4 and Vr=6

-2

-1

0

1

2

-1.5

-1

-0.5

0

0.5

1

1.5

240.5 241 241.5 242 242.5 243 243.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

Fig. 15 Vorticity contours for two cylinders in a tandem arrangement when the cylinders are at the lowest location at L/D=4 and Vr=4, 4.5, 5 and 7

Fig. 16 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=4

-5-4-3-2-1012345

-1

-0.5

0

0.5

1

325 325.5 326 326.5 327 327.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

Fig. 17 Vorticity contours for two cylinders in a tandem arrangement at L/D=6 and Vr=8

-2

-1

0

1

2

-1.5

-1

-0.5

0

0.5

1

1.5

132 132.5 133 133.5 134 134.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a) (b) (c) (d)

Fig. 18 Variations of mean drag and root mean square lift coefficients with L/D for two cylinders in the side-by-side arrangement at Re=150

0

0.5

1

1.5

2

1 2 3 4 5 6

CD

L/D

Cylinder 1

Cylinder 2

Single cylinder

(a) Mean Drag coefficient

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6

CL

,rm

s

L/D

Cylinder 1Cylinder 2Single cylinder

(b) RMS lift coefficient

Fig. 19 Time histories of the cylinder displacement for the side-by-side arrangement

-0.1

0

0.1

350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500

Y/D

(a) L/D=1.5,Vr=2.5

-1

-0.5

0

0.5

1

250 260 270 280 290 300 310 320 330 340 350

Y/D

(b) L/D=1.5,Vr=5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110

Y/D

(c) L/D=1.5,Vr=15.5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110

Y/D

(d) L/D=1.5,Vr=16

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110

Y/D

(e) L/D=1.5,Vr=20

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80

Y/D

(f) L/D=1.5,Vr=30

-1

-0.5

0

0.5

1

200 210 220 230 240 250 260 270 280 290 300

Y/D

(g) L/D=2,Vr=6.5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90 100 110 120 130 140

Y/D

(h) L/D=2,Vr=13.5

-1

-0.5

0

0.5

1

10 20 30 40 50 60 70 80 90

Y/D

(i) L/D=2,Vr=20

-1

-0.5

0

0.5

1

10 20 30 40 50 60

Y/D

(j) L/D=2,Vr=30

-1

-0.5

0

0.5

1

60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220

Y/D

(k) L/D=4,Vr=3.5

-1

-0.5

0

0.5

1

310 320 330 340 350 360 370 380 390 400 410 420

Y/D

(l) L/D=4,Vr=6.5

-1

-0.5

0

0.5

1

160 170 180 190 200 210 220 230 240 250 260

Y/D

tfn

(m) L/D=6,Vr=4

-1

-0.5

0

0.5

1

190 200 210 220 230 240 250 260

Y/D

tfn

(n) L/D=6,Vr=6

Fig. 20 Contours of the real part of the wavelet transform of the vibration displacement in the time-frequency plane for two cylinders in the side-by-side arrangement

Fig. 21 Variation of the response amplitude with the reduced velocity for the side-by-side arrangement

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

A

Side-by-side, L/D=1.5

Side-by-side, L/D=2

Single cylinder

(a) Amplitude for L/D=1.5 and 2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A

Vr

Side-by-side, L/D=4

Side-by-side, L/D=6

Single cylinder

(b) Amplitude for L/D=4 and 6

Fig. 22 FFT spectra of the displacement and the lift coefficient for two cylinders in the side-by-side arrangement

0

0.5

1

1.5

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(a) L/D=1.5, Vr=3.5

0

0.5

1

1.5

0

0.5

1

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(b) L/D=1.5, Vr=5

0

0.5

1

1.5

0

0.5

1

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(c) L/D=1.5, Vr=8.5

0

0.2

0.4

0.6

0

0.5

1

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(d) L/D=1.5, Vr=12

0

0.1

0.2

0

0.1

0.2

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(e) L/D=1.5, Vr=18

0

1

2

3

0

0.2

0.4

0.6

0.8

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(f) L/D=2, Vr=4

0

0.1

0.2

0.3

0

0.2

0.4

0.6

0 1 2 3

AC

L

Ay/

D

f/fn

Y

CL

(g) L/D=2, Vr=10

0

0.1

0.2

0

0.1

0 1 2 3 4

AC

L

Ay/

D

f/fn

Y

CL

(h) L/D=2, Vr=15

0

0.1

0.2

0

0.1

0.2

0 1 2 3 4

AC

L

Ay/

D

f/fn

Y

CL

(a) L/D=2, Vr=25

Fig. 23 Variation of the response frequency with the reduced velocity for the side-by-side arrangement

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Side-by-side, L/D=1.5

Side-by-side, L/D=2

Single cylinder

(a) Frequency for L/D=1.5 and 2

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f/fn

Vr

Side-by-side, L/D=4

Side-by-side, L/D=6

Single cylinder

(b) Frequency for L/D=4 and 6

Fig. 24 RMS lift coefficient on the two cylinders in the side-by-side arrangement

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CL

,rm

s

Side-by-side L/D=1.5

Side-by-side L/D=2

Single cylinder

(a) CL,rms of Cylinder 1 for L/D=1.5 and 2

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Side-by-side L/D=4

Side-by-side L/D=6

Single cylinder

(b) CL,rms of Cylinder 1 for L/D=4 and 6

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CL

,rm

s

Side-by-side L/D=1.5

Side-by-side L/D=2

Single cylinder

(c) CL,rms of Cylinder 2 for L/D=1.5 and 2

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Side-by-side L/D=4

Side-by-side L/D=6

Single cylinder

(d) CL,rms of Cylinder 2 for L/D=4 and 6

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CL

,rm

s

Vr

Side-by-side L/D=1.5

Side-by-side L/D=2

Single cylinder

(e) CL,rms of total force for L/D=1.5 and 2

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CL

,rm

s

Vr

Side-by-side L/D=4

Side-by-side L/D=6

Single cylinder

(f) CL,rms of total force for L/D=4 and 6

Fig. 25 Variation of the phase between the response displacement and the total lift force with the reduced velocity for two cylinders in the side-by-side arrangement

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)Side-by-side, L/D=1.5

Side-by-side, L/D=2

(a) Phase between Y and CL for L/D=1.5 and 2

-45

0

45

90

135

180

225

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ψ(d

egre

e)

Vr

Side-by-side, L/D=4

Side-by-side, L/D=6

(b) Phase between Y and CL for L/D=4 and 6

Fig. 26 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5 and Vr=8.5

-4

-3

-2

-1

0

1

2

3

4

-1.5

-1

-0.5

0

0.5

1

1.5

248 248.2 248.4 248.6 248.8 249 249.2 249.4 249.6 249.8 250

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a)(b) (c) (d) (e) (f)

Fig. 27 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=1.5 and Vr=25.

-2

-1

0

1

2

-1

-0.5

0

0.5

1

52 52.5 53 53.5 54 54.5 55 55.5 56

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2(a)(b)(c)(d)(e)(f) (g) (j)(h) (i)

Fig. 28 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=2 and Vr=4.5

-3

-2

-1

0

1

2

3

-1

-0.5

0

0.5

1

319 319.2 319.4 319.6 319.8 320 320.2 320.4 320.6 320.8 321

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d) (e) (f)

Fig. 29 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=4 and Vr=6

-1

0

1

-1

-0.5

0

0.5

1

481.5 482 482.5 483 483.5 484 484.5

CL

CL

,rm

s

tfn

Y/D CL1 CL2 CL1+CL2

(a) (b) (c) (d)

Fig. 30 Vorticity contours for two cylinders in a side-by-side arrangement at L/D=6, Vr=3 and L/D=4, Vr=3

-1

0

1

0 50 100 150 200 250

Y/D

tfn

(c) Time histories for L/D=6, Vr=3. Initial disturbation is introduced.